I wish there was a way to set a custom icon for the system notifications. Is there a way?
Nyet...
The macOS doesn't allow developers to monkey with notifications.
You could do your own custom notifications using the Custom HTML Prompt action if you've got the chops...
You might want to check this utility out:
You could take a look at DEPNotify -- Notifications will still have the DEPNotify icon (as Chris says, Apple doesn't allow you to avoid that), but you can include a second image using the NotificationImage
command.
While primarily developed for device enrolment workflows and not much use for gathering user input, DEPNotify is also good for alerts, progress indicators, and similar.
It's a packaged application, not an installer, and it looks like you've launched it with the package defaults?
Have Terminal ready, open the package again, and in Terminal type
echo "Status: Hello World" >> /var/tmp/depnotify.log
...and you'll see the bottom line of text change. Then
echo "Command: Quit: Thanks for testing me" >> /var/tmp/depnotify.log
...to Quit with a dialog.
Thinking about it, I've never seen DEPNotify used without the main window showing and I can't immediately see a way of seeing notifications without the window being visible -- closest I can get is to use AS to set the window position so it has just a small part on screen. So probably a non-starter if you just want to throw up Notifications.
Apologies for the bum steer...